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#4340. divisibility rules for the sum of odd numbered consective in

Mathematics, level: Senior
Posted Wed Mar 11 02:56:05 PDT 2009 by Randall Hudson (Randall Hudson).
Huntsville High School, Huntsville, U.S.A.
Activity Time: 30
Concepts Taught: Divisibility rules/deductive proofs/consecutive integer problems

After your students have been working consecutive integer problems for a while you can spring this lesson on them. For them to try to form a deductive proof that the sum of an odd number of consecutive integers is divisible by the number of integers. For instance, that the sum of three consecutive integers is divisible by three. First you can adopt an inductive approach. List 10, 11, and 12 and ask the students if they notice anything about their sum. Hopefully, someone will say, 'It's divisible by three.' Then you can list 5, 6, 7, 8, and 9 and ask if they notice anything about their sum. Hopefully, someone will say, 'It's divisible by 5'. They should have caught on to the concept by now, if not give them another example or two. Now jump to the deductive proof side and ask them to come up with a deductive proof that the sum of three consecutive integers is divisible by three. n+n+1+n+2= 3n+3 which is clearly divisible by three. Just to make sure they understand ask them to come up with a deductive proof that the sum of five consecutive integers is divisible by five. n+n+1+n+2+n+3+n+4=5n+10. Now, ask the students to form a deductive proof that the sum of an even number of consecutive integers is not divisible by the number of them. For instance, have the students proove that the sum of two consecutive integers is not divisible by two. n+n+1=2n+1 (clearly not divisible by two) and that the sum of four consecutive integers is not divisible by four. It should be easy for them by now: n+n+1+n+2+n+3=4n+6 which clearly is not divisible by 4. This lesson should reinforce consecutive integer concepts and give students some practice at forming simple deductive proofs. Good luck and good teaching.